MATH 536B: Algebraic Geometry (Spring 2019)

Essential course information is contained in the syllabus.
A summary is below.

Instructor

Jack Hall
email: firstnamelastname at math dot arizona dot edu
office: ENR2 S349

Text

Various.

Class

when: Tuesday and Thursday, 12:30pm-1:45pm
where: MTL 124.

Office hours

when: Friday, 12-1pm.
where: ENR2 S349.

Topics covered

1/17: Presheaves, sheaves, pushforward, inverse image, sheafification.
1/29: Stalks, sheaves of modules, quasi-coherent and coherent sheaves.
1/31: Pullbacks of sheaves of modules, Quasi-coherent sheaves on affine schemes.
2/5: Quasi-coherent sheaves on affine schemes (continued), schemes, open immersions, topological properties of schemes, reduced schemes.
2/7: Integral schemes, affine morphisms, pushforward and pullback of quasi-coherent sheaves, immersions.
2/12: Closed immersions and quasi-coherent sheaves of ideals, images, separated and universally closed morphisms.
2/14: Proper morphisms, valuation rings, and valuative criteria.
2/19: Valuative criteria (continued), quasi-coherent sheaves on projective schemes.
2/21: Quasi-coherent sheaves on projective schemes (continued).
2/26: Ampleness, very ampleness, and associated primes of modules over rings.
2/28: Associated points on schemes and the sheaf of meromorphic functions.
3/12: Cartier divisors, Weil divisors.
3/14: Weil divisors (continued).
3/15: (extra class) Cohomology.
3/19: Cohomology of ringed spaces.
3/21: Cohomology of quasi-coherent modules schemes and Serre vanishing.
3/22: (extra class) Cohomology of projective space.
3/26: Coherence of higher pushforwards along proper morphisms of locally noetherian schemes, towards Riemann's Theorem.
3/28: Riemann's Theorem, Hilbert polynomials, and flatness.
3/29: (extra class) Flat limits.
4/2: Formally smooth, etale, and unramified morphisms.
4/4: Kahler differentials.
4/5: (extra class) Deformation theory of smooth morphisms of schemes.
4/9: Fundamental exact sequences, characterizations of unramified morphisms.
4/11: Euler exact sequence and applications.
4/12: (extra class) Riemann--Roch and applications.
4/16: Classification of curves of low genus.
4/18: Classification of curves of low genus (continued).
4/19: (extra class) Riemann--Hurwitz.
4/23: Hyperelliptic curves, the canonical embedding and Serre duality.
4/25: Serre duality (continued).